Understanding Pair of Angles in Geometry

When two lines share a common endpoint (often called a vertex), the angles present on either side of the vertex can be called the pair of angles. Now, these are pretty important when it comes from a geometrical point of view because pairs of angles have a set of similar properties. These properties can be used to find the unknown angle among a bunch of known angles. This article explains various types of paired angles in detail. 

Pair of Angles

Complementary Angles

Any two pairs of angles having a common vertex, giving a sum as 90° or a right angle (like an L), are referred to as complementary angles.

Between the 2 angles present, one is called the ‘complement’ of the other.

A Complementary Angle

Here, $\angle B O C=90^{\circ}$

Also, $\angle B O A=60^{\circ}$ and $\angle A O C=30^{\circ}$ and

$\angle B O A$ and $\angle A O C$ are complements of each other and complementary angles.

Supplementary Angles

Any two pairs of angles having a common vertex, giving a sum as $180^{\circ}$, or a straight angle are referred to as supplementary angles.

Between the 2 angles present, one is called the ‘supplement’ of the other.

A Supplementary Angle

Here, $\angle A O C=180^{\circ}$

Also, $\angle B O C=50^{\circ}$ and $\angle B O A=130^{\circ}$ and

$\angle B O A$ and $\angle B O C$ are supplements of each other and supplementary angles.

Adjacent Angles

Two angles having one common arm, one common vertex, and the other two arms lying on the opposite side of this common arm so that their interiors do not overlap are known as adjacent angles.

Adjacent Angles

Vertically Opposite Angles

When two lines intersect at a common point, 4 angles are formed surrounding the point of intersection. Opposite angles having no common arm are called vertically opposite angles. Vertically opposite angles are equal.

Vertically Opposite Angles

Here, Lines $A B$ and $C D$ intersect at point $O$, as vertically opposite angles are equal.

$\angle \mathrm{AOC}=\angle \mathrm{DOB}=35^{\circ}$

$\angle A O B=\angle D O C=145^{\circ}$

Thus, vertically opposite angles are equal.

Linear Pair

Two angles whose sum is 180° can be defined as a linear pair. These angles have a common arm and a common vertex.

A Linear Pair

Here, $\angle 1+\angle 3=180^{\circ}$

Hence, $\angle 1$ and $\angle 3$ form a linear pair.

Solved Examples

1. If angles $\angle A$ and $\angle B$ are supplementary angles, $\angle A$ is found to be $46^{\circ}$. Find $\angle B$.

Ans: Here, $\angle A$ and $\angle B$ are supplementary angles.

$\angle A+\angle B=180^{\circ}$

$46^{\circ}+\angle B=180^{\circ}$

$\angle B=180^{\circ}-46^{\circ}$

$\angle B=134^{\circ}$

Thus, we find $\angle B$ to be $134^{\circ}$.

Practice Questions

1. $\angle \mathrm{A}$ and $\angle \mathrm{B}$ are complementary angles. $\angle \mathrm{B}$ is $72^{\circ}$, find $\angle \mathrm{A}.$ 

Ans: $18^{\circ}$

2. $\angle \mathrm{X}$ and $\angle \mathrm{Y}$ form a linear pair. $\angle \mathrm{X}$ is $123^{\circ}$, find $\angle \mathrm{Y}.$ 

Ans: $57^{\circ}$

3. $\angle Q$ and $\angle R$ are supplementary angles. $\angle Q$ is $143^{\circ}$, find $\angle R.$ 

Ans: $37^{\circ}$

Summary

In the given article, we discussed pairs of angles, including linear pairs of angles, and vertically opposite angles. Then we talked about a pair of complementary angles and examples and then we discussed a pair of complementary angles. These pairs of angles are of various types and are an important part of geometry. Complementary angles are those angles whose sum is 90°. Supplementary angles are those angles whose sum is 180°. Adjacent angles are angles sharing a common arm and a common vertex; they also have an opposite arm which produces the non-overlapping angle. Vertically opposite angles are formed when two lines intersect producing 4 angles surrounding it. Angles that are opposite, with no common arm, fall under this category. Vertically opposite angles are equal to each other.